In the paper we examine the problem of optimization of parameters of technological process for obtaining a molten pool with a needed depth on the surface of the metal sample under the thermal local laser treatment. A mathematical model of heat distribution during thermal laser treatment is simplified to a linear problem of heating a half-space, that can be used for plates of finite thickness. Despite a number of assumptions, this approach gives satisfactory qualitative and quantitative understanding of the heating stage. The accepted model allows us to control the quality criteria of the surface layer and, above all, its main feature - the depth of the molten pool by selecting the optimal parameters of laser radiation. To solve the optimization problem the algorithm based on the scanning method of the search space (the zero-order method of conditional optimization) has been built. The obtained results can be used both as preliminary recommendations for setting variable parameters of the real technological process and as prediction of phase and structural states of the irradiated material.
In the modern industry the laser treatment of the materials has already become quite common. Such processes as laser cutting, welding, punching, marking, etc. have been widely used. The application of fiber lasers is the most effective, due to their small size, ease of use and the possibility of pervasion into difficult accessible places [1–3]. In addition, the fiber lasers of the new generation are of high quality of optical radiation, of high power and high speed of the scanning on the material surface. Modern laser technologies are complicated and depend on many factors. The same technological process can not proceed under the absolutely same conditions. Therefore, the development of Copyright c by the paper’s authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org a set of recommendations for the specific equipment to ensure successful achievement of the technological process results is an important task.
One of the popular laser thermal processing (laser scanning on the material surface) is the formation of a molten pool with different configurations and sizes in the surface layer of the sample [3]. Thermal material parameters (density, heat capacity, thermal conductivity, thermal diffusivity, melting temperature, boiling temperature, etc.), the geometric sizes of the sample, and the technological possibilities of the laser (lower and upper power limits of the laser treatment, the minimum and maximum spot radius of the laser radiation, etc.) affect the outcome of the process. Also an important factor is the action time (the exposure time of laser radiation). Experimental selection of variable parameters for obtaining a molten pool of a required size is a laborious, expensive and sometimes impossible process. Therefore, mathematical modeling and solving optimization problems, which arise in the solution of this issue, are always relevant.
In this paper the problem of choosing the optimal parameters of the thermal laser treatment for obtaining a molten pool with geometric sizes, which are the closest to the required, is examined. The calculations are carried out in the framework of the mathematical model of heat distribution under the local heating of the material (structural steel A) by laser treatment. The source of laser treatment is considered to be an impulse one. The source treatment power varies in the range from 20 to 1000 watts, the treatment time – from 0.01 to 0.1 sec, the laser spot radius of 0.0025 to 0.005 meters. Since the increasing of the laser source power and the treatment time lead to growing the depth of the molten pool and uncontrolled rising in the temperature on the surface of material until its boiling and vaporization, one of the main constraints on the process result is the condition that the material must not boil at the surface of the molten pool. 1
Conducting technological operation is determined by the specific impact of the laser
radiation on the material (depends on the selected device) and specified features of
a technological problem. The power, wavelength and time of radiation exposure are
usually considered as the main parameters [4] among many characteristics of laser
radiation. Here we examine the process of thermal action of the radiation, heating the
object to a specific temperature, therefore the laser radiation power P is assumed as
the main characteristic. The pulse power and mean power are considered for pulsed
lasers. The mean power depends on the pulse time and frequency. The working object
is heated to the temperature T , which is determined by the power density Q of
absorbed radiation. The distribution of Q on the irradiated surface S can be specified
in the various forms. For example [5], the absorbed power density Q with the uniform
distribution law is directly proportional to the product of the radiation power P0 by
a coefficient of surface absorption of laser radiation A and inversely proportional to
the square of the irradiated zone S. For calculations a Gaussian distribution law of the
absorbed power density within the irradiated zone usually used most often [6]:
Q = Aqmaxe−κr2 ,
(
LASER 0
R H h Z r0 r
X
Equation (
The boundary condition on the surface Z = 0 specifies the action of surface source [6]: −k ∂T (r, Z, t) ∂Z
Z=0 = (1 − Kref )q(r, t), r 6 r0, 0, r > r0, where q(r, t) – the radiation power density on the body surface, Kref – the surface reflection coefficient for the metal in the zone of influence of the laser radiation (1 − Kref = A), (1 − Kref )q(r, t) – the absorbed specific radiation flux, k – the thermal conductivity of the material.
The solution of the problem (
∂T (r, Z, t)|r=∞ = ∂T (r, Z, t)|Z=∞ = ∂T (r, Z, t)|t=0 = T0
allows to specify the relationship between the temperature T (r, Z, t) of the irradiated
material with the power density of the laser radiation q(r, t). In (
Assuming impulse heating, when r0 ≫ √aτ , a solution of (
Z = 0.
Formula (
Ar0 arctan k√π 2√at r0 −1
(
On the surface of the irradiated material (r 6= 0, Z = 0) the solution of problem
(
At the depth of the irradiated material (r 6= 0, Z 6= 0) we obtained the solution of
problem (
T (r, Z, t) =
qmaxe−r2/r02 Erfc T (r, Z, t) =
qmaxe−r2/r02 Erfc 2A√at
k 2A√at k
Z 2√at
The relations (
T (P0, r0, t, r, Z) = 2A√at k Here the problem of choosing the optimal parameters of a technological process for creating the molten pool by pulsed laser thermal treatment is formulated. For this purpose a formalism similar to [8–10] is used. The laser treatment power P0, the spot radius r0 and the action time t of the laser radiation are taken as variable parameters. The vector of varied parameters is defined as
x = (x1, x2, x3), where x1 = P0, x2 = r0, x3 = t. The variable parameters (14) are limited by the inequalities
ximin 6 xi 6 ximax, i = 1, 3.
In (15) the minimum and maximum values for each parameter xi must be set by taking into account the technical capabilities of the equipment. In this paper the characteristics of industrial fiber laser with a maximum power of 1000 W [7] are selected for modeling.
Output parameters characterizing the creation process of the molten pool are denoted as
y(x) = (y1, y2, y3),
where y1 = Tc(x) – the temperature in the spot center on the plate surface (calculated
from (
1) the material temperature in the center of the laser spot on the sample surface must be greater than the melting point and less then the boiling point:
Tf < Tc(x) < Tb; 2) the radius of the molten pool on the sample surface must be greater than zero and less than the radius of the laser spot: 0 < R(x) < r0; 0 < H(x) < H∗.
3) the depth of of the molten pool must be greater than zero and less than the target value H∗:
The problem is to choose the parameters x so that the feasibility conditions of the technological process would be satisfied and the depth of the molten pool H(x) would be close to a given value H∗.
Using the notation adopted above, the statement of the problem can be written in the form [8–10]:
x = arg xm∈Dinx |y3(x) − H∗|, where the allowable set Dx in the space of variable parameters, for which the constraints (15) and the conditions of feasibility of the process (17) are satisfied, has the form Dx = x | xmin 6 x 6 xmax, ymin < y(x) < ymax . (19)
Thus, the solving of the problem (18)–(19) lies in searching such set of variable parameters x = (x1, x2, x3), at which the minimum of the objective function ΔH(x) is achieved:
ΔH(x) = |y3(x) − H∗| .
For solving the optimization problem with constraints of a special kind [11] a scanning method of the search space (the zero-order conditional optimization) on a discrete set of values x and y(x) is used. Step dxi for each of the variable parameters xi is assumed constant. The solution algorithm is presented in block diagram form in Fig. 2. The use of first or second-order optimization methods in this case is rather difficult because of the complex form of model relations.
BLOCK 1 performs the calculation of the discrete set of the output parameters yj (x) (j = 1, 2, 3) at each grid point (x1, x2, x3), at the same time the conditions of successful feasibility of technological process (17) are checked for each yj . If all conditions (17) are met, the current point of space x and the corresponding y are suitable and recorded to the row n in the two-dimensional array {x; y} (n – counter of suitable iterations).
BLOCK 2, separately allocated on the scheme as part of BLOCK 1, contains the evaluation of the values y3 = H(x), given in implicit form by the equation T (x, H(x)) = Tf , which can be solved with respect to H by dichotomy method.
In the BLOCK 3, the values ΔH = y3k(xk) − H∗ are calculated for the all saved sets {xk; yk} (k = 1, 2, ...n). Then, the minimum is determined among all of ΔH, and (18) (20) FALSE
x2=x2+dx2; x3=x3min Tm < y1 < Tb
TRUE
Tth=Tm
y2=R(x1,x2,x3) y2min < y2 < y2max y3=H(x1,x2,x3)
x1=x1min; x2=x2min; x3=x3min; n=0 x1 <= x1max
FALSE Record=( k : min abs(y3k-H*), k=1,2..n ) Print(xRecord; yRecord)
END
the corresponding set number is saved in the variable ”Record”. Thus, the solution of the problem (18)–(19) is a set of variable parameters xRecord, delivering a minimum of the objective function (20). 3
For the computational experiment the necessary thermal characteristics of the chosen material (Table 1) and laser processing parameters (Table 2) are specified. The characteristics of the material (structural steel A) are defined according to [12]. The parameters in Table 2 correspond to industry ytterbium fiber lasers [7].
The solution of the choice problem of an optimum set of technological process parameters is obtained for the given material characteristics (Table 1) and the laser treatment parameters (Table 2). The values dxi (i = 1, 2, 3), allowing us to achieve the desired accuracy in the calculation of the objective function, are taken as the following: dx1 = 20, dx2 = 0.00025, dx3 = 0.01. At the defined limit of the molten pool depth H∗ = 0.0005 m the allowable set Dx consists of 197 vectors x = (x1, x2, x3) which satisfy the feasibility conditions of the process (17). The Table 3 shows a part of the resulting discrete set of suitable collections of variable and output parameters. It is obvious that the solution of the optimization problem for a given value of H∗ is a vector x1 = (960; 0.00275; 0.08) (P0 = 960 W, r0 = 0.00275 m and t = 0.08 s), minimizing the objective function ΔH(x).
Additionally to the shown above solutions, the algorithm presented in this paper is used for calculating the maximum depth of a molten pool which satisfies all the necessary conditions and can be obtained on the surface of the selected material (Table 1) by the laser with given specifications (Table 2). For this purpose the value H∗ in (20) is taken equal to the sample thickness h = 0.004 m. The obtained isothermal diagram of the molten pool with the maximum possible depth H = 0.000978 m is shown in Fig. 3 and corresponds to the solution x = (960; 0.0025; 0.1) (P0 = 960 W, r0 = 0.0025 m and t = 0.1 s).
To obtain a deeper molten pool it is necessary to increase the maximum values of variable parameters of the laser. However, this method, firstly, is not always technically possible, and secondly, may cause an unwanted effect – boiling of the material at the sample surface.
Thus, the paper offers a version of solving the problem of choosing the optimal process parameters of laser thermal treatment to create the molten metal pool with a predetermined depth. The obtained results can be used as a pre-theoretical foundation for experimental testing of real laser system. The calculating algorithm takes into account the basic restrictions on the technical parameters of the laser and the necessary conditions for realization of the process.
Acknowledgments. The work is supported by Ministry of Education and Science of the Russian Federation (Decree #P218, Agreement #02.G25.31.0116 on 14.08.2014 0.0
0.001 2500 2000 1500 Tm 1000 Fig. 3. The diagram of temperature distribution across the processed sample height. The dark area limited by dashed isotherm Tm corresponds to the molten pool with a depth H and radius R on the sample surface between JSC ”Ship repairing center ”Dalzavod” and Ministry of Education and Science of the Russian Federation).